[Paper Review] Tropical Atmospheric Circulations with Humidity Effects

Chun-Hsiung Hsia, Chang‐Shou Lin+2 more|arXiv (Cornell University)|Oct 18, 2011

Advanced Thermodynamics and Statistical Mechanics9 references3 citations

TL;DR

This paper studies the impact of atmospheric humidity on tropical large-scale circulation using a Boussinesq model coupled with a diffusive humidity equation and linearized moisture-dependent heating. It demonstrates via dynamic transition theory that humidity slightly reduces the critical thermal Rayleigh number for convection onset, preserving the same Type-I (continuous) transition and El Niño-like oscillatory mechanism as in dry models, confirming the robustness of the self-organizing ENSO feedback system under moist conditions.

ABSTRACT

The main objective of this article is to study the effect of the moisture on the planetary scale atmospheric circulation over the tropics. The modeling we adopt is the Boussinesq equations coupled with a diffusive equation of humidity and the humidity dependent heat source is modeled by a linear approximation of the humidity. The rigorous mathematical analysis is carried out using the dynamic transition theory. In particular, we obtain the same types of transitions and hence the scenario of the El Niño mechanism as described in \cite{MW2,MW3}. The effect of the moisture only lowers slightly the magnitude of the critical thermal Rayleigh number.

Motivation & Objective

• To investigate the effect of atmospheric humidity on planetary-scale tropical atmospheric circulation.
• To determine whether moisture alters the type of dynamic transition (e.g., continuous vs. jump) in large-scale tropical convection.
• To assess whether the El Niño–Southern Oscillation (ENSO) mechanism, previously derived in dry models, remains valid under moist conditions.
• To quantify the influence of humidity on the critical thermal Rayleigh number for convection onset.
• To validate the robustness of the self-organizing, self-excitation ENSO mechanism under inclusion of moisture effects.

Proposed method

• Uses the Boussinesq approximation for atmospheric motion, incorporating a diffusive equation for water vapor mixing ratio.
• Models the latent heat release as a linear function of humidity, representing the moisture-dependent heating source.
• Applies dynamic transition theory to analyze the stability and transition behavior of the system near critical parameters.
• Solves the eigenvalue problem for the linearized system to determine the critical Rayleigh number and convection patterns.
• Derives the critical thermal Rayleigh number formula including humidity effects, showing its dependence on the dimensionless humidity Rayleigh number.
• Performs asymptotic analysis and numerical approximation using realistic atmospheric parameters (e.g., Prandtl number, diffusivity, tropospheric height).

Experimental results

Research questions

• RQ1 How does atmospheric humidity affect the onset of large-scale tropical atmospheric convection?
• RQ2 Does the inclusion of moisture change the type of dynamic transition (e.g., from continuous to jump) in the tropical circulation model?
• RQ3 What is the quantitative impact of humidity on the critical thermal Rayleigh number for convection onset?
• RQ4 Does the El Niño-like oscillatory mechanism persist when humidity is included in the model?
• RQ5 How do convection scales (e.g., wave number, length scale) compare with observed Walker circulation patterns under moist conditions?

Key findings

• The presence of humidity slightly reduces the critical thermal Rayleigh number for dynamic transition, as the coefficient of the humidity Rayleigh number in the critical Rayleigh formula is negative.
• The system undergoes only continuous (Type-I) dynamic transition under idealized boundary conditions, consistent with the dry model and the ENSO mechanism.
• The critical Rayleigh number is approximated as $ R_c \simeq \delta_1 \pi^2 = 2.76 \times 10^{21} $, which aligns with values from dry models.
• The convection length scale is estimated at $ L_c \simeq 3350 \, \text{km} $, consistent with the observed scale of the tropical Walker circulation.
• The wave number at criticality is $ k_c \simeq 6 $, corresponding to a zonal wavelength of about 3350 km, matching large-scale tropical atmospheric patterns.
• The critical temperature difference for convection onset is $ \Delta T_c = 60^\circ \text{C} $, which is consistent with the dry model and physically plausible.